Problem: Let $f(x) = 2x^{2}-6x-5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $2x^{2}-6x-5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 2, b = -6, c = -5$ $ x = \dfrac{+ 6 \pm \sqrt{(-6)^{2} - 4 \cdot 2 \cdot -5}}{2 \cdot 2}$ $ x = \dfrac{6 \pm \sqrt{76}}{4}$ $ x = \dfrac{6 \pm 2\sqrt{19}}{4}$ $x =\dfrac{3 \pm \sqrt{19}}{2}$